The 'Download' is a recorded copy of the show, also know as a Podcast.

What you are listening to on your laptop IS the Podcast of those shows.
BBC World Service - Find A Programme - Assignment

If you are just clicking on Listen, you are hearing a streaming download of the podcast. It's streaming and not saving to be listened to later. If you want to listen to that show later, you need to download the podcast (MP3) and move it to your device.

Again I recommend a program like HappyFish. After you set it up it automaticly downloads the latest files from your subscriptions. Then you can synchronize (copy to) using the same software.
HappyFish >}};> Add New Feed
HappyFish >}};> Sync Setup


A Podcast is just a term generally meaning audio other than music.
Most Podcasts are a talk radio type format and usually distributed in MP3 audio codec. Other commom audio formats you may see are, .AAC, .WMA, and .WAV.